Tài liệu Hiệu suất của hệ thống thông tin máy tính P10 pptx

We first introduce basic terminology in Section 10.1 after which we discuss the class of feed-forward queueing networks in Section 10.2.. Although Jackson queueing networks can be applie

Part III

ISBNs: 0-471-97228-2 (Hardback); 0-470-84192-3 (Electronic)

Chapter 10

systems consist of multiple, fairly independent service providing entities, each with their own queue Jobs in such systems “travel” from queueing station to queueing station in

places: in computer systems where a number of users are trying to get things done by

a set of processors and peripherals, or in communication systems where packets travel

Chapter 4 we already saw an example of a queueing network: the simple terminal model

in which many system users attended a central processing system and their terminals in a

limited

We first introduce basic terminology in Section 10.1 after which we discuss the class

of feed-forward queueing networks in Section 10.2 This discussion provides us with a good insight into the analysis of more general open queueing networks, such as Jackson networks in Section 10.3 Although Jackson queueing networks can be applied in many cases, there are situations in which the model class supported does not suffice, in particular when arrival streams are not Poisson and service times are not exponentially distributed Therefore, we present in Section 10.4 an approximation procedure for large open queueing networks with characteristics that go well beyond the class of Jackson networks We finally

study in Section 10.5

Performance of Computer Communication Systems: A Model-Based Approach.

Boudewijn R Haverkort Copyright © 1998 John Wiley & Sons Ltd ISBNs: 0-471-97228-2 (Hardback); 0-470-84192-3 (Electronic)

10.1 Basic terminology

Queueing networks (QNs) consist of a number of interconnected queueing stations, which

we will number 1, s , M Individual queueing stations or nodes, are independent of one another They are, however, connected to each other so that the input stream of customers

of one node is formed by the superposition of the output streams of one or more other nodes,

We assume that there is a never-empty source from which customers originate and arrive

at the QN, and into which they disappear after having received their service

A more formal way to describe QNs is as a directed graph of which the nodes are the queueing stations and the vertices the routes along which customers may be routed from node to node The vertices may be labelled with routing probabilities or arrival rates When dealing with open QNs, the source (and sink) is generally denoted as a special node, mostly numbered 0 (we will do so as well)

A well-known example of an open QN model is a model of a (public) telecommunication

We will address such a model in Section 10.5

10.2 Feed-forward queueing networks

In this section we discuss feed-forward queueing networks (FFQNs) In such networks the queues can be ordered in such a way that whenever customers flow from queue i to queue

j, this implies that i < j, i.e., these QNs are acyclic Note that due to this property FFQNs must be open We focus on the case where the individual queueing stations are of MIMI 1 type, but will indicate generalisations to multi-server queueing stations

steps We first discuss the MIMI1 q ueue as the simplest case of an FFQN in Section 10.2.1

We then discuss series of MIMI1 queues in Section 10.2.2 and finally come to the most general form of FFQNs in Section 10.2.3

10.2.1 The MIMI1 queue

Consider a simple MIMI1 queue with arrival rate X and service rate p Given that the queue operates in a stable fashion, i.e., p = X/p < 1, we know from Chapter 4 that the steady-state probability of having i customers in the queue is given by

10.2 Feed-forward queueing networks 201

Figure 10.1: Series connection of M queues

The correctness of this result can be verified by substituting it in the global balance equa-

CTMC to be equal to JV

10.2.2 Series of MIMI1 queues

Now consider the case where we deal with M queues in series The external arrival rate

to queue 1 equals X and the service rate of queue i equals pi For stability we require that all pi = X/pi < 1 If there are queues for which pj 2 1, these queues build up an infinitely large waiting line and the average response time of the series of queues goes to infinity as

link) of the QN In Figure 10.1 we show a series QN

Since there are no departures from the QN, nor arrivals to the QN in between any two queues, the arrival rate at any queue i equals X Furthermore, a departure at queue

the state space Z = LV 2 Every state (i, j) E Z signifies the situation with i customers in queue 1 and j customers in queue 2 The sum of i and j is the total number of customers in the series QN As can be observed form Figure 10.2, a “column of states” represent states with the same overall number of customers present in the QN Recognising that there are basically 4 different types of states, it is easy to write down the GBEs for this CTMC:

Solving these GBEs seems a problem at first sight; however, their regular structure and

/ -

P2

/ / / -

Figure 10.2: CTMC underlying two MIMI 1 queues in series

P&j = (1 - Pl)P”; x (1 - P2,d (10.3)

is the correct solution That this is indeed the case, we leave as an exercise to the reader Now we will spend some more time on interpreting this result In a series QN it seems that

For series QNs the product-form property is easy to show In fact, all the customer streams

in a series QN are Poisson streams This fact has been proven by Burke and is known as:

The departure process from a stable single server MIMI 1 queue with arrival and

That Burke’s theorem is indeed valid, can easily be seen Consider an MIMI 1 queue

as sketched As long as the queue is non-empty, customers will depart with the inter-

10.2 Feed-forward queueing networks 203

departure the queue empties, one has to wait for the next arrival, which takes a negative exponentially distributed length (with rate X), plus the successive service period So, when leaving behind an empty queue, the time until the next departure has a hypo-exponential

upon departure instances the queue is non-empty equals p, so that we can compute the

which reduces to Fo (t) = 1 - epxt = FA (t)

In fact, Burke’s theorem also applies to MIM(m queues, in which case the output process

of every server is a Poisson process with rate X However, it is easy to see that for an MIMI 1 queue with feedback, the resulting outgoing job stream is not a Poisson stream any more

In fact, also the total stream of jobs entering such a queue is no Poisson stream The superposition of the external arrival stream and the jobs fed back after having received service is not a Poisson process since the superpositioned Poisson streams are no longer independent

individual MIMI1 q ueue simply apply As an example, the average number of customers

in queue 2 is derived as

E[N,] = gyjPi,j = p& -P1)Pi x 7ll - Pa)Pi

ix0 j=o i=o j=o

Z = LVM and find the following steady-state probabilities:

i=l

(10.6)

where G is called the normalising constant, i.e.,

which assures that the sum of all probabilities CnEZ Pr{N = n} = 1

10.2.3 Feed-forward queueing networks

In this section we address general FFQNs, i.e.: acyclic QNs, not necessarily in series All

overall arrival process from the environment be a Poisson process with rate X0 Let ri,j

definition, we have ri,j = 0 whenever j 5 i The probabilities ro,+ indicate how the arrivals

the QN

The overall flow of jobs through queue j now equals the sum of what comes in from the environment and what comes from other queues upon service completion, i.e.,

number of queues in the QN As FFQNs are acyclic we can solve the traffic equations successively:

NOW, if all the pi = Xi/pi < 1, the QN is said to be stable If that is the case, the

probabilities are given as:

10.3 Jackson queueing networks 205

When we deal with multiple server FFQNs, a similar results hold Only the normali- sation constant changes according to the results derived in Chapter 4 for MlMlm queues

In any case, Burke’s theorem applies so that in FFQNs all the job streams are Poisson streams

10.3 Jackson queueing networks

Jackson QNs (JQN s ) are an extension of FFQNs in the sense that the “feed-forward restric- tion” is removed, i.e., jobs may be routed to queues they attended before The job streams between the various queues now are not Poisson streams because they are composed out

of dependent streams Surprisingly enough, however, the queue-wise decomposition of the

QN can still be applied! Despite the fact that the job streams are not Poisson, the steady- state probabilities for the QN take a form as if they are ! This also implies that we still

with the FFQN is now given by the traffic equations:

(10.11)

These cannot be solved successively any more, so that we either have to use a Gaussian elimination procedure or an iterative technique (see Chapter 15) Once the values Xi are known we can establish the stability of the QN by verifying whether pi = Xi/pi < 1 for all

i If this is indeed the case, we have the following solution for the steady-state customer probability distribution:

As we will see in the application in Section 10.5, we do not always have to specify

queueing network and to compute the values Xj directly from them Furthermore, under the restriction that all queues are stable, the throughputs Xi are equal to the arrival rates

Xi, for all queues

Figure 10.3: A simple Jackson QN distributions for the multiple-server queueing stations (all according to the results given in Section 4.5) The first-order traffic equations remain unchanged

Consider the simple 2-station Jackson QN given in Figure 10.3 We denote the external arrival rate at queue i as X~ro,i; the overall arrival rate at queue i is Xi Services in queue i take place with rate pi and after service, the customer leaves the QN with probability ri,o or goes to the other queue with probability ri,3-i = 1-ri,a The corresponding state-transition diagram is given in Figure 10.4 From the state transition diagram we can conclude the following GBEs:

Po,#o + i-42) = p 0,~ 1 0 - X r 0,2 + Po,j+1rcL2r2,0 + pl,jwl,0 + pl,j4w1,2, j E JV + , P&O + PI + p2) = pi+ I 0 0,2 + Pi+l,j-lwl,2 X r + pi+l,jw1,0

+ pi,j+lp2r2,0 + pi-l,j-lp2r2,1 + pi l,jXor0,1, i, j E N+ (10.13)

To verify whether the general solution given satisfies these GBEs (including the normali- sation equation) we have to solve the traffic equations first The overall traffic arriving at queue i can be computed as:

1 equals that what flows in from the environment, plus that which flows in after it has

10.4 The QNA method 207

xoro,1 w-l,0 x07-0,2

I-Lzr2,o

w-l,2 1-12r2,l

Figure 10.4: State transition diagram for the simple Jackson QN

been at queue 2 However, since customers may cycle through the queues more than once, this sum has to be “stretched” by a factor (1 - r1,2r2,1)-1

The stability conditions are p; = Xi/pi < 1 Using these pi-values, the result for Jackson

10.4 The QNA method

In many modelling applications, the restriction to Poisson arrival processes and negative

an example, think of a model for a multimedia communication system in which arrivals

of fixed-length packets (containing digitized voice or video) have either a very bursty or

a very deterministic nature For such systems, Jackson queueing networks might not be the most appropriate modelling tool Instead, a modelling method known as the Queueing

quick analyses of large open queueing networks with fixed routing probabilities and FCFS scheduling Its most important characteristic, though, is that it treats queueing networks

characterised by their first two moments, i.e., successive arrivals are still independent of

moments; in particular, constant services times can be dealt with, albeit approximately

well as for multiple customer classes In its simplest form, it reduces to an exact analysis

of Jackson queueing networks as we have seen in Section 10.3

ing stations Once the traffic equations have been solved, all queueing stations can be stud-

the overall queueing network into a number of individual queueing stations is approximate

We will discuss how good this approximation is

This section is further organised as follows The considered queueing network class is introduced in section 10.4.1 The computational method is discussed in Section 10.4.2 and

a summary of the involved approximation steps is given in Section 10.4.3

10.4.1 The QNA queueing network class

tomers travel between nodes according to fixed routing probabilities ri,j, i, j = 0, 1, , M (; and j not both equal to 0) There is one special node, the environment (indexed 0) from which external arrivals and to which departures take place

bination (merging) so as to represent the segmentation and reassembly process in com-

following parameters have to be defined:

i.e., by the arrival rate X o,i and the squared coefficient of variation C~;o,i;

10.4 The QNA method 209

l the number of servers mi;

l the service time distribution, characterised by the first) and second moment, i.e., by the average service time E[Si] = l/pi and the squared coefficient of variation C~;i;

l the routing probabilities ri,j;

a the customer creation or combination factor Ti, with

for combination stations, for ordinary stations, (1, oo), for creation stations

(10.18)

For all the queues, it is assumed that the scheduling strategy is FCFS and that the buffer

is infinitely large

10.4.2 The QNA method

The general approach in the QNA method consists of four steps:

tion 10.4.1;

generation and the solution of the traffic equations, both for the first moment (similar

to that seen in Section 10.3), and for the second moment (to be discussed below);

the first and second moment of the service time distribution, and the first and second moment of the interarrival time distribution, using exact and approximate results for

throughput, the customer departure rate, mean and variance of the sojourn time and number of customers, per queue and for the overall network

These steps are treated in more detail in the following subsections

Input

to a single class queueing network by aggregating customer classes

Once all the parameters for the single class queueing network have been specified, it might be the case that some values ri,i > 0, i.e., some queues have immediate feedback

the immediately fed-back customer as one larger visit, and to compensate for this later

follows:

ci;i + ri,i + (1 - ri,i)Cg;i)

ri,j rii + 1 _ ri,i ' Li # 4

When computing the performance measure per node, we have to correct for these changes

In this step the customer flows between nodes should be obtained We first concentrate

on the mean traffic flow, i.e., the arrival rates Xi to all the nodes The first-order traffic equations are well-known and given by

Xj = &,j + 5 &Tiri,j, j = l;**,M, (10.20)

i=l

that is, the arrival rate at node j is just the sum of the external arrival rate at that node, and the departure rates X; of every node Xi, weighted by the customer creation factor yi and the appropriate routing probabilities ri,j

There are basically three operations that affect the traffic through the QN and which are illustrated in Figure 10.5:

(a) the probabilistic splitting of a renewal stream, induced by the constant routing prob- abilities which take place after customer completion at a queueing station;

10.4 The QNA method 211

Figure 10.5: The basic operations (a) splitting, (b) superpositioning and (c) servicing that affect the traffic streams

(b) the service process at a particular queueing station;

(c) the superpositioning of renewal streams before entering a particular queueing station The fact that the first-order traffic equations are so easily established comes from the fact that the superpositioning of renewal streams as well as the probabilistic splitting of renewal streams can be expressed as additions and multiplications of rates Also, the service process

at a queueing station does not affect the average flows

The first-order traffic equations form a non-homogeneous system of linear equations, which can easily be solved with such techniques as Gaussian elimination or Gauss-Seidel iterations

Now that the arrival rates to the queues have been found, it is possible to calculate the utilisation pi = Xi/mipi at node i If pi 2 1 for some i, that queueing station is overloaded

If all the pi < l9 the queueing network is stable and the analysis can be taken further

To determine the second moment of the traffic flows we use the so-called second-order traffic equations The three operations that affect the traffic characteristics are superpositioning and probabilistic splitting of renewal streams, and the service process We therefore focus

on these three factors and the way they influence the second-order characteristics of the arrival processes

As we have seen in Chapter 3, if a renewal stream with rate X and squared coefficient

of variation C2 is split by independent probabilities oi, i = 1, , n, then the out- going processes are again renewal processes, with rates Xi = aiX and with squared coefficient of variation (2’: = aiC2 + (1 - ai) If, however, at node i customer creation

or combination takes place with value yi, then C2 is scaled by a factor yi as well, i.e., Cf = oiyiC2 + (1 - ai) It should be noted that this scaling is an approximation

(b) Servicing The service process also has its influence on the departure process If the service process has a very high variability, it increases the variability of the outgoing stream, or on the other hand, a more deterministic service process decreases the variability of the outgoing stream (in comparison with the variability of the incoming stream)

Apart from the service time variability, the utilisation of the server is also of impor- tance, and therefore the average service time, i.e.,

- when the utilisation is low, the outgoing stream will more closely resemble the

characteristics of the incoming stream;

- if the utilisation is high, there will almost always be customers present so that

the departure process is almost completely determined by the service process

In QNA , an adaptation of Marshall’s result for Ci, the variability of the departure process in the GIlGIl q ueue, is used Marshall’s result states that

We observe that the coefficient of variation of the service process is weighted with p2

and the coefficient of variation of the arrival process is weighted with (1 - p2) to add

up to the coefficient of variation of the departure process

For multiple server queues, i.e., GIIGI m q ueues, this result has been extended to

G = 1+(1-P2)( c; - 1)$&p; - 1)

- ~c~+(l-P”)c2,+P2(l-~j